Have Lunch on the Beach, See the Earth Curve!

Not every day do you get an amazing science lesson during lunch.

Most days, I eat lunch at the computer, or I take a sandwich in the car to eat while driving errands. Then come the glorious exceptions. A cooler on the beach. My wife next to me. 

Some time back, the sun was hot, the air cool and clear. We found a beach log to sit on just ten feet from the lapping waves. Turkey sandwiches, chips, apples, perfection. Okay, I forgot cookies, so it wasn't perfect. But close. The water was indigo, and the mountains – Mt. Tallac especially – still had some small snowfields in the high bowls to the side of The Cross.

The cerulean sky was decorated with jet contrails from travelers who probably looked down and thought, “Oh, my God, look at that view! Why are we flying someplace else when we could have gone to Tahoe?!”

Our postprandial activity was sailboat watching through binoculars. There were a variety of boats transforming wind into movement and play. The most interesting one was across the lake about 10 or 12 miles, over by Cave Rock, just a flicker to the naked eye, but easy to watch in the binoculars. It had a tall mast and a beautiful sail curved into a graceful, power-generating airfoil shape.

But what made it striking in looks was that the sail had no boat.

WHAT? NO BOAT?!

That's what we saw. A sail going back and forth with no hull below.

That's how much the earth curves.

Tahoe as seen from SR-71 Blackbird from approximately 90,000 feet
Photo credit F-16.net

It was fascinating to watch, this mast and sail dancing the waves sans boat.
Back home, I did some research. 
The simplest thing to say about the earth's shape is that it curves about 8 inches per mile. It would seem, then, that a 6-foot-tall person could see the water's surface about 9 miles away (6 feet = 72 inches. Divide by 8 inches – the amount of curve per mile – and you get 9 miles.)

Unfortunately, I learned that it ain't that simple.

The first complication in assessing long distance curvature is that with each additional mile, the earth's surface is curving away from you at 
an ever-increasing angle. Because of this, a 6-foot-tall person can see the water's surface only about 3 miles away.

If you want some techy explanation, go here:

http://mathcentral.uregina.ca/qq/database/QQ.09.02/shirley3.html

This ever-increasing characteristic makes a huge difference as the distances increase.

If you want to see the waterline 10 miles away, you'd have to be about 66 feet up in the air.

If you want to see the waterline 22 miles away (the length of Tahoe), you'd have to be about 300 feet up in the air.

Now comes the second main complication. Because of the cool, denser air near the cold water's surface, light curves toward that surface just like it curves when it is refracted through a lens. The amount of refraction varies with different temperatures and other climatic conditions, but it can be substantial.

The result of this refraction negating the effects of curvature is that if you stand on Tahoe's North Shore, you may be able to sometimes see the tall hotels of the South Shore. But sometimes you won't because they are technically “below the curve of the earth” as seen from the North Shore. Only when light from them bends are they visible.

Of course, there is always one more thing to keep in mind. The earth's curvature is only noticeable when you are right down on the water. Most of the land around Tahoe is substantially above the lake. If you are up the mountain slope just a few hundred feet – as many of us are most of the time – then nothing on the shore is “below the curve of the earth.”

So next time you picnic at the water's edge, sit down close to the water and watch the sailboats with binoculars. You may be in for a science display treat!
P.S. Remember to bring cookies.